Finite Group Databasebeta

← #22597983

Subgroups — #22597983

Order 9088.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×D71C211yesmaximal, characteristic, complemented
2C2×#14556134C211yesmaximal, characteristic, complemented
3#17986565C212yesmaximal, complemented
4#17986251C212yesmaximal, complemented
5#17986250C212yesmaximal, complemented
6#17986236C211yesmaximal, characteristic, complemented
7#17986235C211yesmaximal, characteristic, complemented
8#17986232C211yesmaximal, characteristic, complemented
9#17986036C211yesmaximal, characteristic, complemented
10#17986002C212yesmaximal, complemented
11#466×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×D4×D71C2212yes—
14C2×D568C2211yescharacteristic
15C2×#11765850C2212yes—
16D8×D71—24no—
17#14556134—24no—
18C2×#11765849C2211yescharacteristic, complemented
19C2×#11765848C2211yescharacteristic
20C2×#11765848C2211yescharacteristic, complemented
21C2×C8×D71C2211yescharacteristic
22C2×C8×D71C2211yescharacteristic, complemented
23#14556124—22no—
24#14556114—22no—
25#14556103C2212yescomplemented
26#14556102C2212yescomplemented
27#14556099C2212yescomplemented
28#14556098—22no—
29#14556096—22no—
30#14556086C2212yes—
31#14556062C2211yescharacteristic, complemented
32#14556059C2212yescomplemented
33#14556050C2212yescomplemented
34#14556049C2212yescomplemented
35#14556048C2212yescomplemented
36C2×D8×C71C2211yescharacteristic
37C22×C8×C71C2211yescharacteristic, complemented
38#121×C71C2212yescomplemented
39#107×C71C2211yescharacteristic, complemented
40#102×C71C2212yescomplemented
41C23×D71—22no—
42#11765865—22no—
43#11765865—22no—
44C2×Dic142D411yescharacteristic
45C2×D284—22no—
46C2×D284D411yescharacteristic
47C2×D284C2312yes—
48C2×C22⋊D71—22no—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71D411yescharacteristic
52C2×C22⋊D71D411yescharacteristic
53D4×D71—24no—
54D4×D71—24no—
55C2×C4×D71C2311yescharacteristic
56C2×C4×D71C2311yescharacteristic
57D568—22no—
58#11765850—24no—
59#11765849—22no—
60#11765848—22no—
61#11765848—22no—
62C2×C71⋊C8C2311yescharacteristic
63C2×C71⋊C8C2311yescharacteristic
64C8×D71—22no—
65C8×D71D414yescomplemented
66C4⋊Dic71C2312yes—
67C22⋊Dic71—22no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—22no—
71C4×Dic71—22no—
72C2×D4×C71—22no—
73C2×D4×C71C2312yes—
74C22×C4×C71C2311yescharacteristic
75D8×C71—22no—
76C2×C8×C71—22no—
77C2×C8×C71C2311yescharacteristic
78C2×C8×C71C2311yescharacteristic
79C4⋊C4×C71C2312yes—
80C22⋊C4×C71—22no—
81C23×C71—22no—
82C23×C71C2×D411yescharacteristic
83C22×D71—22no—
84C22×D71—44no—
85C22×D71—82no—
86C22×D71—82no—
87C22×D71C2×D411yescharacteristic
88C22×D71C2×D411yescharacteristic
89D4×C71—22no—
90D4×C71—22no—
91D4×C71—82no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—22no—
98C22⋊D71—22no—
99C22⋊D71—44no—
100C22⋊D71—82no—
101C22⋊D71—82no—
102C2×Dic71—22no—
103C2×Dic71C2×D411yescharacteristic
104C2×Dic71C2×D411yescharacteristic
105D284—22no—
106D284—22no—
107D284—22no—
108D284—82no—
109C4×D71—21no—
110C4×D71—21no—
111C4×D71C2×D412yes—
112C4×D71#4112yes—
113Dic142—22no—
114C8×C71—21no—
115C8×C71C2×D412yes—
116C71⋊C8—21no—
117C71⋊C8C2×D412yes—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—42no—
122C22×C71—42no—
123C22×C71—42no—
124C22×C71C22×D411yescharacteristic
125C2×D71—21no—
126C2×D71—21no—
127C2×D71—21no—
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—88no—
134C2×D71#12112yes—
135C4×C71—21no—
136C4×C71—42no—
137C4×C71C22×D411yescharacteristic
138C4×C71C2×#4111yescharacteristic
139Dic71—21no—
140Dic71—42no—
141Dic71#12112yes—
142C2×C71—41no—
143C2×C71—82no—
144C2×C71C2×#12111yescharacteristic
145C2×C71#57611yescharacteristic
146C2×C71#57611yescharacteristic
147D71—22no—
148D71—41no—
149D71—82no—
150#3058—711nomaximal, Sylow
151C71#305811yescharacteristic, complemented, Sylow
152C22×D8—711no—
153C2×#131—711no—
154#535—712no—
155#469—712no—
156#469—712no—
157#466—711no—
158#466—711no—
159#466—711no—
160#466D7111yescharacteristic, complemented
161#427—711no—
162#418—712no—
163C2×#41—711no—
164C22×D4—712no—
165C2×D8—712no—
166C2×D8—711no—
167C2×D8—1424no—
168C2×D8C2×D7111yescharacteristic
169#131—1424no—
170C2×M16—711no—
171C2×M16—711no—
172C2×M16—711no—
173C22×C8—711no—
174C22×C8—711no—
175C22×C8C2×D7111yescharacteristic, complemented
176#127—1422no—
177#124—1422no—
178#121—712no—
179#121—712no—
180#121—712no—
181#121C2×D7112yescomplemented
182#120—1422no—
183#120—1422no—
184#117—712no—
185#107—711no—
186#107C2×D7111yescharacteristic, complemented
187#106—712no—
188#102—712no—
189#102—712no—
190#102—712no—
191#102C2×D7112yescomplemented
192C24—1422no—
193#41—1422no—
194#41—1422no—
195C2×Q8—711no—
196C2×D4—22no—
197C2×D4—711no—
198C2×D4—711no—
199C2×D4—711no—
200C2×D4—712no—
201C2×D4—1422no—
202C2×D4—1424no—
203C2×D4—1424no—
204C2×D4—1422no—
205C2×D4—1422no—
206C2×D4—1422no—
207C2×D4C22×D7112yes—
208C22×C4—711no—
209C22×C4—711no—
210C22×C4C22×D7111yescharacteristic
211D8—22no—
212D8—1424no—
213D8—1422no—
214M16—1422no—
215M16—1422no—
216M16—1422no—
217C2×C8—22no—
218C2×C8—711no—
219C2×C8—714no—
220C2×C8—711no—
221C2×C8—1422no—
222C2×C8C22×D7111yescharacteristic
223C2×C8C22×D7111yescharacteristic
224C4⋊C4—712no—
225C4⋊C4C22×D7112yes—
226C22⋊C4—22no—
227C22⋊C4—1422no—
228C22⋊C4—1422no—
229C22⋊C4—1422no—
230C22⋊C4—1422no—
231C42—1422no—
232C23—22no—
233C23—2844no—
234C23—5682no—
235C23—5682no—
236C23—711no—
237C23—711no—
238C23—1422no—
239C23D4×D7111yescharacteristic
240Q8—1422no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—2844no—
245D4—5682no—
246D4—5682no—
247D4—5682no—
248D4—1422no—
249D4—1422no—
250D4—1422no—
251D4—1422no—
252D4—1422no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—711no—
257C2×C4—712no—
258C2×C4—712no—
259C2×C4—711no—
260C2×C4—1422no—
261C2×C4—1421no—
262C2×C4—1421no—
263C2×C4D4×D7111yescharacteristic
264C2×C4C23×D7111yescharacteristic
265C8—21no—
266C8—712no—
267C8—1421no—
268C8D4×D7112yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—2842no—
276C22—2842no—
277C22—2842no—
278C22—5688no—
279C22—712no—
280C22—1421no—
281C22—1421no—
282C22—1421no—
283C22—1421no—
284C22—1421no—
285C22C2×D4×D7111yescharacteristic
286C4—21no—
287C4—42no—
288C4—2842no—
289C4—712no—
290C4—1421no—
291C4C2×D4×D7111yescharacteristic
292C4#41×D7111yescharacteristic
293C2—41no—
294C2—82no—
295C2—2841no—
296C2—5682no—
297C2—1422no—
298C2#121×D7111yescharacteristic
299C2#1798680011yescharacteristic
300C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#466×C71C2
socleC22×C71C22×D4
hypercenter#466D71
hypocenterC71#3058
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D8×D71C2semidirect
2. C2×#14556134C2semidirect
3. #17986565C2semidirect
4. #17986251C2semidirect
5. #17986250C2semidirect
6. #17986236C2semidirect
7. #17986235C2semidirect
8. #17986232C2semidirect
9. #17986036C2semidirect
10. #17986002C2semidirect
11. #466×C71C2semidirect
18. C2×#11765849C22semidirect
20. C2×#11765848C22semidirect
22. C2×C8×D71C22semidirect
25. #14556103C22semidirect
26. #14556102C22semidirect
27. #14556099C22semidirect
31. #14556062C22semidirect
32. #14556059C22semidirect
33. #14556050C22semidirect
34. #14556049C22semidirect
35. #14556048C22semidirect
37. C22×C8×C71C22semidirect
38. #121×C71C22semidirect
39. #107×C71C22semidirect
40. #102×C71C22semidirect
65. C8×D71D4semidirect
151. C71#3058semidirect
160. #466D71semidirect
175. C22×C8C2×D71semidirect
181. #121C2×D71semidirect
186. #107C2×D71semidirect
191. #102C2×D71semidirect